We use mathematical models to study the mechanisms of oscillatory electrical activity arising from ion channels in cell membranes and modulated by intracellular chemical processes. We are interested in both the behavior of single cells and the ways in which cells communicate and modify each other's behavior. Our main application has been to the biophysical basis of insulin secretion in pancreatic beta-cells. We have examined bursting oscillations in membrane potential and the role of electrical coupling between cells in the islet of Langerhans. Long term goals are to understand how the membrane dynamics interact with intracellular events to regulate secretion. We also compare, contrast, and generalize to other secretory cells and neurons, including GnRH-secreting hypothalamic neurons, pituitary somatotrophs, and fast neurotransmitter secretion at nerve terminals. Our primary tool is the numerical solution of ordinary and partial differential equations. We use analytical, geometrical, graphical, and numerical techniques from the mathematical theory of dynamical systems to help construct and interpret the models. Perturbation techniques are used to get analytical results in special cases. We study both detailed biophysical models and simplified models which are more amenable to analysis. Such an approach aids the isolation of the essential or minimal mechanisms underlying phenomena, the search for general principles, and the application of concepts and analogies from other fields. Another role for our group is to mediate between the mathematical and biological disciplines. This includes disseminating the insights of mathematical work to biologists in accessible language and alerting mathematicians and other theoreticians to new and challenging problems arising from biological issues. Recent work on this project includes: 1. (Islet calcium and voltage oscillations) We have applied our recently developed calcium subspace model to illuminate the differences between single beta cells and islets. In particular, we showed that the detailed properties of isolated cells can be accounted for if channel noise is included in the model. The three classes of single-cell behavior we observe (spikers, fast bursters, and plateau cells) are then obtained in the model by varying calcium channel conductance. This single parameter change explains the decrease in spike amplitude, decrease in frequency, and increase in plateau fraction as one progresses through the three classes. In a complementary study, we have (with the experimental laboratory of L. Satin) contrasted two hypotheses for the role of gap junctional coupling. One possibility is that individual beta-cells are capable of islet-like oscillations, and the coupling is needed only to synchronize the oscillations. Alternatively, it may be that coupling is needed for oscillations to occur at all. We have tested this by using anti-sense mRNA for the gap junction protein connexin 43 (Cx43) to reduce coupling strength. We find that islets with reduced coupling behave like the single cells in our previous study: they show fast spiking or bursting, but not the slow bursting seen in intact islets. With R. Bertram, we have analyzed the dynamics of the current class of models, which include ER calcium dynamics and oscillatory ATP/ADP ratio. We trace the development of such models from the earliest beta-cell model, showing the contribution of each included mechanism. Inclusion of ER dynamics is sufficient to account for the increase of burst frequency in the presence of the insulin-secretion potentiator acetylcholine. Inclusion of nucleotide ratio dynamics permits for the first time simulation of the triphasic transient response of islets to a step of glucose (latency, first phase spiking, and steady-state oscillation). We have also explored a model in which negative feedback is provided not by internal calcium, but by autocrine and paracrine effects of secreted insulin. 2.(Computer modeling of calcium diffusion and buffering) We have applied our CalC ("Calcium Calculator"; http://mrb.niddk.nih.gov/matveev) software package for simulation of buffered Ca2+ diffusion in a presynaptic terminal, to explore the buffer saturation hypothesis proposed by E. Neher. This hypothesis attempts to explain short-term synaptic facilitation as a consequence of an increase in the amplitude of successive calcium spikes due to saturation of endogenous buffers. This contrasts with a model we have previously published in which a rise of the residual calcium remaining after the spikes is responsible for facilitation. The two models are not mutually exclusive, but can coexist. It is also possible that different mechanisms predominate in different types of nerve terminals. We have carried out a systematic analysis of the conditions for the two mechanisms to operate. 3. (Metabolic insulin signaling) The detailed model of metabolic insulin signaling, which we previously developed, has now appeared. The computer files are posted at http://mrb.niddk.nih.gov/sherman/insulin.html in xpp format and have also been included in a CellML repository: http://www.cellml.org/examples/repository/sedaghat_model_2002_doc.html.